I think you're out of luck here. What I mean by that is that I don't think there is a formula of the type that you are looking for.

Think about the particular case of the product that arises when . There is then only one term in the product, and it is equal to . For most values of n, there is no explicit formula for in terms of more elementary functions of n. So even in that special case, there is no satisfactory solution to the problem.

Originally Posted by wglmb

I've found this identity: , which looks useful but isn't quite what I need.

The fact that there is a good solution in this case is due to the fact that the numbers are the roots of a fairly easily computable polynomial of degree n–1, and the product of the roots is given by the constant term in the polynomial. But that doesn't apply if you only have a subset of the roots.